Image Description using Radial Associated Laguerre Moments

Transcription

1 J. ICT Res. Appl., Vol. 9, No., 25, -9 Image Descriptio usig Radial Associated Laguerre Momets Boju Pa, Yihog Li 2 & Hogqig Zhu 3* College of Egieerig, Northeaster Uiversity, Bosto, Massachusetts 25, USA 2 Techology Ceter, AVIC Shaaxi Baocheg Aviatio Istrumet Co., Ltd, Baoji, Shaaxi 726, Chia 3 School of Iformatio Sciece & Egieerig, East Chia Uiversity of Sciece ad Techology, Shaghai 2237, Chia pa.bo@husky.eu.edu Abstract. This study proposes a ew set of momet fuctios for describig gray-level ad color images based o the associated Laguerre polyomials, which are orthogoal over the whole right-half plae. Moreover, the mathematical frameworks of radial associated Laguerre momets (RALMs) ad associated rotatio ivariats are itroduced. The proposed radial Laguerre ivariats retai the basic form of disc-based momets, such as Zerike momets (ZMs), pseudo-zerike momets (PZMs), Fourier-Melli momets (OFMMs), ad so o. Therefore, the rotatio ivariats of RALMs ca be easily obtaied. I additio, the study exteds the proposed momets ad ivariats defied i a gray-level image to a color image usig the algebra of quaterio to avoid losig some sigificat color iformatio. Fially, the paper verifies the feature descriptio capacities of the proposed momet fuctio i terms of image recostructio ad ivariat patter recogitio accuracy. Experimetal results cofirmed that the associated Laguerre momets (ALMs) perform better tha orthogoal OFMMs i both oise-free ad oisy coditios. Keywords: associated laguerre polyomials; orthogoal laguerre momets; quaterio; radial-polar; rotatio ivariats. BItroductio Orthogoal momet fuctios are oe of the most importat tools for the costructio of shape descriptors for patter recogitio [], object classificatio [2], template matchig [3], image recostructio [4], ad data compressio [5], etc. Amog all kids of momets, cotiuous ZMs, PZMs, OFMMs ad Legedre momets were itroduced by Teague ad Teh, et al. i refs. [6]-[9] accordig to their correspodig polyomials as kerel fuctios. As is kow to all, the computatio of ZMs, PZMs ad OFMMs requires the trasformatio from image coordiates to a regio withi the uit circle, ad Legedre momets eed to trasform image coordiates ito the iterval [-, ]. I additio, discretizatio of the cotiuous itegrals is ecessary for the Received September 4 th, 24, Revised March 7 th, 25, Accepted for publicatio April 29 th, 25. Copyright 25 Published by ITB Joural Publisher, ISSN: , DOI:.564/itbj.ict.res.appl

2 2 Boju Pa, et al. computatio of all cotiuous orthogoal momets. Discrete orthogoal momets provide a more accurate descriptio for image features by evaluatig momet compoets directly i the image coordiate space []-[3]. Hece, the discrete orthogoal momets elimiate the abovemetioed problems associated with the cotiuous momets by usig discrete orthogoal polyomials as kerel fuctios. Recetly, the problem of ivariat patter recogitio which udergoes geometric trasforms, such as rotatio, scalig ad traslatio (RST), has received icreased iterest i the patter recogitio field. Ivariace feature selectio always plays a importat role i this subject. I recet decades, a umber of orthogoal momets-based methods have bee reported to costruct a ivariace feature [4]-[5]. Owig to the polar coordiate represetatio of the kerel fuctios of disc-based orthogoal momets, these cotiuous orthogoal momets have better image feature represetatio capabilities. This is maily due to the fact that their rotatio ivariats ca be easily costructed. Hece, as far as patter recogitio tasks is cocered, the cotiuous orthogoal momets still outperform discrete momets. Borrowig the specificity of the radial polyomials of disc-based momets, Mukuda [6] recetly itroduced the framework of radial Tchebichef momets, which is based o the structure of disc-based momets ad is particularly suitable for patter recogitio works requirig rotatio ivariats. I this paper, we propose aother kid of discrete orthogoal momets, called associated Laguerre momets, which are also useful for image descriptio. The proposed ALMs are defied i terms of the associated Laguerre polyomials [7], which are orthogoal over the whole right-half plae. The advatage of the proposed ALMs over disc-based cotiuous orthogoal momets lies i the fact that the computatio of these cotiuous momets requires a coordiate trasformatio ad suitable approximatio of the itegrals that are ot existet i the proposed ALMs. Takig cues from Mukuda s research works [6], this study combies the merit of computatioal advatages of discrete orthogoal momets with shape descriptio capabilities of cotiuous orthogoal momets, ad itroduces RALMs i polar-coordiate form. Thus, the shape descriptors of rotatio ivariats, which retai the basic form of disc-based cotiuous momets, ca be easily costructed i terms of the magitude of the proposed RALMs. Sice the scale ad traslatio ivariace of a patter ca be achieved by other methods, such as Fourier methods [8], Melli methods [9], Rado trasform methods [2] ad ormalized methods [2] etc., this paper omits their discussio due to limited space. Furthermore, the proposed method obtais Q- ALMs ad Q-RALMs by extedig ALMs ad RALMs to the quaterio field, ad derives a set of rotatio ivariats based o them i order to deal with color images by usig quaterio algebra. The advatage of this type of

3 Image Descriptio usig Radial Associated Laguerre Momets 3 represetatio is that a color image ca be treated as a vector field. The accuracy of the proposed ALMs, RALMs, Q-ALMs, ad Q-RALMs as global feature descriptio capabilities is assessed by meas of image recostructio i oise-free ad oisy cases ad the results are compared with those of OFMMs. A classificatio experimet o gray-level ad color images also illustrates the usefuless of the proposed feature descriptors. The remaider of this paper is orgaized as follows: Sectio 2 gives some mathematical backgroud o the associated Laguerre polyomials ad quaterio algebra theory. Sectio 3 presets a set of discrete ALMs ad RALMs, ad ivestigates the rotatioal ivariace of RALMs. I Sectio 4 presets the defiitio of Q-ALMs ad Q-RALMs i a holistic maer. Sectio 5 presets the experimetal results ad illustrates the performace of the proposed shape descriptors, ad Sectio 6 summarizes the paper. 2 Backgroud This sectio reviews the associated Laguerre polyomials ad quaterio algebra, ad presets some properties that will be useful i the rest of the paper. 2. Associated Laguerre Polyomials It is well kow that the associated Laguerre polyomials { L } x are orthogoal with respect to the weight fuctio ( ) x < +, that is, ( ) wx, for > -, = xe o the iteral x e x L ( x) Lm ( x) dx = δ m,, m () Γ + +! where δ m is Kroecker s symbol. The associated Laguerre polyomials are defied as ( + ) L ( x) = F( ; + ; x) (2)! where the Pochhammer symbol () k is as follows ( ) = ( + )( + 2) ( + k ) with ( ) ad F ( x) k = (3) ; + ; is a cofluet hypergeometric fuctio of the first kid ( + ) ( ) k = ( a) ( ) 2 k a a a z k z F( abz ; ; ) = + z+ + = (4) b bb+ 2! b k! k

4 4 Boju Pa, et al. Together with Eqs. (4) ad (2), the associated Laguerre polyomials ca be rewritte as k ( + )! k L ( x) = ( ) x (5) ( k )!( k + )! k! k = The associated Laguerre polyomials satisfy the followig secod-order recurrece relatio: ( ) ( ) ( ) ( ) ( ) xl x = + L x x L x (6) Quaterio Algebra A quaterio has four compoets (oe real part ad three imagiary parts) ad ca be represeted i a hyper-complex form as [22] q= a+ bi + c j+ dk (7) where abcd,,, R, ad i, j, k obey the followig multiplicatio rules: i = j = k = i j = j i = k,, j k = k j = i, k i = i k = j The modulus of a quaterio q follows the defiitio for complex umbers as q = a + b + c + d (9) It is ofte useful to cosider a quaterio as the sum of a scalar part ad a vector part, which is represeted as q= S( q) + V( q) () where S(q) = a ad V(q) = b i+c j+d k. If S(q) =, the q is reduced to a pure quaterio. 2.3 Associated Laguerre Momets Without loss of geerality, the ALMs of a image f(x, y) with order of m+ ad size of N N are defied by the ormalized associated Laguerre orthogoal poyomials L ( x) as follows N N m = m x= y= S L ( xl ) ( y) f( xy, ), m, =,,,N-. () (8)

5 Image Descriptio usig Radial Associated Laguerre Momets 5 ( ) Cosiderig the set of associate Laguerre polyomials, { L } is ot suitable for defiig momets because the rage of values of the polyomials expads rapidly with the icrease of order. To avoid umerical fluctuatio i the momet computatio, the curret study applies ormalized associated orthogoal Laguerre polyomials L ( x) ( + k) to defie the proposed ALMs. x xe! L ( x) = L ( x)! (2) The first few orders of the ormalized associated Laguerre polyomials with the parameters =,, 2 are show i Figure. From Figure oe ca observe clearly that the values of ormalized associated polyomials L ( x) are bouded o a fiite iterval ad have a otable differece from the associated L x. polyomials ( ) (a) Figure Plot the L ( x) ad L ( x), (a) = ; (b) = ; (c) = 2.

6 6 Boju Pa, et al. (b) (c) Figure Cotiued. Plot the L ( x) ad L ( x), (a) = ; (b) = ; (c) = 2.

7 Image Descriptio usig Radial Associated Laguerre Momets 7 Comparig Eq. () ad Eq. (2), oe ca obtai the orthogoality coditio of ormalized associated polyomials L ( x) as ( ) ( ) m m L x L x dx = δ, m, (3) The performace of momet descriptor S m is well assessed by meas of image recostructio. Thaks to the orthogoality ad completeess of { L ( x) }, which allow oe to represet ay square itegrable image f(x, y) via a trucated series defied over whole right-half plae, to be writte by N N ( ) ( ) ( y) fˆ xy, = S L xl (4) m= = m m It should be expected that the represetatio i the above formula ca coverge to the true image by makig orders N sufficietly large. 2.4 Radial associated Laguerre Momets Rotatioal ivariace is a iheret property of disc-based cotiuous orthogoal momets. However, as idicated i Eq. (), the proposed ALMs are defied over the Cartesia coordiates, therefore, it is still ot coveiet eough to geerate rotatio ivariats. Motivated ad aided by the framework of Zerike radial polyomials, this study therefore tries to defie the followig RALMs of order p ad repetitio q as R L r e f(, r θ ) (5) m 2π jqθ = p ( ) 2π r= θ = where the image has a size of N N pixels ad m deotes N/2. Sice θ is a real quatity measured i radias, oe ca rewrite (5) as R L r e f(, r θ ) (6) m jqθ = p ( ) r= θ = where is at 36. Whe the image is sampled at oe-degree itervals ad the coordiates x, y are give by rn 2πθ 2 cos N, rn πθ x = + y = si + N 2( m ) 2 2( m ) 2 (7) the structure of the RALMs is very similar to that of disc-based momets. It ca also be easily foud that the defiitio i Eq. (6) yields a set of momets that is orthogoal i the discrete polar coordiate space of the image. Moreover, the mai purpose of this type of represetatio is that the rotatio ivariats ca be

8 8 Boju Pa, et al. easily derived. Give a image f(r,θ), after rotatio by a agle, the image is f(r,θ+φ). Oe has 2π ˆ jq( θ+ φ) R = rl p() r f (, r θ φ) e drdθ 2π + 2π jqφ jqθ = e rl p() r f (, r θ) e drdθ 2π jqφ = e R After applyig ormal operatios, we have ˆ φ jqφ jqφ = = = (8) R e R e R R (9) Comparig Eq. (8) with Eq. (9), it is ot difficult to see that R is with rotatioal ivariace. The correspodig iverse momets trasform is give by the followig equatio: P ( θ ) ( ) Q ˆ jqθ f r, = R L r e (2) p= q= p 3 Quaterio associated Laguerre Momets Let f(, r θ) fri+ fg j+ fbk = fr(, r θ) i+ fg(, r θ) j+ fb(, r θ) k be a RGB image defied i polar coordiates. Therefore, the forward Q-ALMs ca be defied as N N QS = L ( x) L ( y) m m x= y= = 3 N N m x= y= N N ( if + jf + kf ) µ ( ) ( ) N N = L m ( x) L ( y) ( fr + fg + fb) 3 x= y= N N i L m( x) L ( y) ( fg fb) 3 x= y= j 3 k 3 m x= y= L ( x) L ( y) R G B L x L y ( if + jf + kf )( i + j + k) N N m x= y= R G B ( fb fr) L ( x) L ( y) ( fr fg) (2)

9 Image Descriptio usig Radial Associated Laguerre Momets 9 where μ is a uit pure quaterio chose as study. Thus, Eq. (2) ca be rewritte as where i the curret R R R R QS = A + ia + ja + ka (22) R R 2 R 3 m 2 3 R A = S m( fr ) + S m( fg ) + S m( fb ) 3 A A A = = = 3 3 S m( fg ) S m( fb ) S m( fb ) S m( fr ) S 3 ( f ) S ( f m R m G ) µ = ( i+ j+ k)/ 3 (23) As discussed i the above sectio, sice the associated Laguerre polyomials are orthogoal, color images ca be estimated from a fiite umber N of Q- ALMs usig the followig iverse momet trasform: fˆ( x, y ) = N N m= = = 3 QS L ( x) L ( y) N N m= = R R R R 2 3 N N R R R = L m( x) L ( y) ( A + A2 + A3 ) 3 m= = m m N N R R R i L m( x) L ( y) ( A + A2 A3 ) 3 m= = j 3 N N m m= = µ ( x) ( y) ( A + ia + ja + ka ) L L ( i + j + k) ( ) ( y) L x L ( A A + A m R R R 3 N N R R R k L m( x) L ( y) ( A + A A2 ) 3 m= = ) (24) The right-side Q-RALMs of order p with repetitio q are defied as follows

10 Boju Pa, et al. 2 π µ qθ QR = rl () (, ) 2 p r f r θ e drdθ π 2π = rl ( )( )(cos( ) si( )) 2 p r fri fg j fbk qθ µ qθ drdθ π = Re R ( f ) + µ Im R ( f ) i+ Re R ( f ) + µ Im R ( f ) j ( ( R ) ( R )) ( G ) ( G ) ( Re ( R ( fb )) µ Im ( R ( fb ))) k Im ( R ( fr ) ) Im ( R ( fg ) ) + Im ( R ( fb )) + + ( ) ( ) = Re ( R ( fr )) + Im ( R ( fg )) Im ( R ( f )) B i 3 + Re ( R ( fg )) + Im ( R ( fb )) Im ( R ( fr )) j 3 + Re ( R ( fb )) + Im ( R ( fr )) Im ( R ( fg )) k 3 From such a represetatio, the rotatio ivariats are easily achieved by takig the modulus of Q-RALMs. Similar approaches ca be used to obtai left-side Q-RALMs. Owig to limited space we will omit their discussio. 4 Experimetal Results Several experimets were carried out to evaluate the performace of the proposed ALMs ad RALMs. The experimets used six selected test images (show i Figure 2), icludig gray-level images ad color images, with a resolutio of ad 28 28, respectively. (25) (a) Figure 2 Origial test images, (a) gray-level images (size: ); (b) color images (size: 28 28). The mea square error (MSE) was used as the fidelity criteria measurig the resemblace betwee the recostructed images ad the origial oes. It ca be writte as (b)

11 Image Descriptio usig Radial Associated Laguerre Momets ε = f xy f xy (, ) (, ) f ( xy, ) 2 2 (26) where is the stadard Euclidea orm, f(x, y) ad f ˆ( xy, ) respectively represet the origial test ad the recostructed image. 4. Image Recostructio The most effective method to test image descriptio capability of momet fuctios is by image recostructio. By virtue of orthogoal fuctio theory show i Eq. (3), both gray-level ad color images ca be estimated approximately by Eq. (4). Figure 3 shows the recostructio result usig ALMs ad Q-ALMs for various values of the parameter. The secod experimet was carried out to illustrate the image discrimiatio power of the RALMs usig a Chiese character with a size of pixels, as listed i Table. The recostructio results were compared with those usig orthogoal OFMMs. This aalysis was repeated with a gray-level image with a size of pixels, as show i Figure 4. From Table ad Figure 4 we ca fid that the recostructio based o ALMs outperformed the other two types of orthogoal momets. It is observed that the recostructed images usig OFMMs exhibited a peculiar behavior i the viciity of the image ceter. This pheomeo was more promiet for very high orders of momets. (a) (b) (c) Figure 3 Recostructed images usig ALMs ad Q-ALMs for gray-level images ad color images, respectively, (a) = ; (b) = ; (c) = 2.

12 2 Boju Pa, et al. Figure 4 Recostructio of gray-level image usig ALMs ( = 2), RALMs ( = 2) ad OFMMs; (a) origi image; (b) ALMs; (c) RALMs; (d) OFMMs. Table Recostructio usig RALMs with differet m,, P, Q. Image Recostructio Parameter Origial image with size N = 64 RALMs methods ( = 2) m = 32, =384, P max = 3, Q max =, ε = 272 RALMs methods ( = 2) m = 32, =384, P max = 64, Q max = 3, ε = 44 ALMs methods ( = 2) P max = 64, ε = 32 OFMMs methods P max = 64, ε = 39 It is well kow that oise may severely affect the quality of image recostructio. To evaluate the robustess of momets with regards to differet kids of oises, we tested the oise robustess of differet orthogoal momets. Pepper-ad-salt oise at 5% was added to the origial biary image E (see Figure 5). Figure 6 shows the recostructio results usig differet orthogoal momets, ad the correspodig mea square error compariso is depicted i Figure 6(b). Figures 6(a) ad (b) show that the recostructio error of the Eglish letter E usig RALMs was a little smaller tha usig other methods with the icrease of order. (a) Figure 5 The Eglish letter E, (a) letter E without oise; (b) letter E with 5% salt-ad-pepper oise. (b)

13 Image Descriptio usig Radial Associated Laguerre Momets 3 (a) (b) Figure 6 Recostructio error, (a) oise-free case; (b) salt-ad-pepper oise case (5%). 4.2 Rotatio Ivariat Recogitio This subsectio reports a detailed experimetal study o the recogitio accuracy of RALMs i both oise-free ad oisy cases for biary images, graylevel images, ad color images, respectively. Accordig to the defiitio of RALMs, we ca easily kow that the magitude of RALMs remais ivariat uder image rotatio. Thus, they are useful features for rotatio-ivariat patter recogitio. I the curret recogitio task, the followig feature vectors of rotatioal ivariace were chose V = R 2, R 32, R 33, R 4, R 42, R 43, R 44 (27) The Euclidea distace was utilized as the classificatio measure

14 4 Boju Pa, et al. T ( k) ( k) 2 s t = sj tj j= dv (, V ) ( v v ) (28) where V s is the T-dimesioal feature vector for the ukow sample ad V t (k) is the traiig vector for class k. We defied the recogitio accuracy η as Number of correctly classified images η = % (29) The total umber of image used i the test The classificatio experimet was carried out to test the performace of RALM rotatio momets ivariats. A set of similar biary characters with sizes of pixels, show i Figure 7, was used as the traiig set. The experimet was doe for such a character set sice elemets i this set ca be easily misclassified owig to their similarity. Each testig set cosisted of 72 images, which were geerated by rotatig the traiig images to every 5 degrees i the rage of [, 36). The, each image i the traiig set was degraded by 5% pepper-ad-salt oise. Figure 8 shows part of these testig images. The experimets were repeated for each parameter. Table 2 shows the classificatio results usig differet momet ivariats. As ca be observed from this table, RALMs achieved higher recogitio rates. Figure 7 Part of biary characters traiig set. Figure 8 Part of biary characters testig set with 5% pepper-ad-salt oise.

15 Image Descriptio usig Radial Associated Laguerre Momets 5 Table 2 Classificatio results compariso with differet salt-ad-pepper oises. Method Parameter a Noise-free 5% 8% % 5% 2% 8% 72% 74% 64% 52% 56% 2 % % 94% 92% 76% 66% 4 % 92% 78% 74% 66% 6% RALMs 6 % 84% 7% 68% 52% 48% 8 % 84% 66% 66% 54% 46% % 66% 56% 52% 48% 42% 5 % 98% 92% 84% 76% 54% 2 % 86% 66% 5% 4% 26% OFMMs % 86% 78% 72% 6% 62% Figure 9 Part of gray-level objects traiig set. Figure Part of gray-level objects testig set with % pepper-ad-salt oise.

16 6 Boju Pa, et al. To illustrate the discrimiatio power of RALMs for oisy gray-level images several gray-level traiig images with a size of pixels from the Columbia object image database [23] were chose (show i Figure 9). A ew set of 72 images was geerated as the testig set by rotatig the traiig images from to 36 with a iterval of 5. This was followed by addig saltad-pepper oise with differet oise desities, as show i Figure. I the classificatio work, seve ivariats of RALMs ad OFMMs were calculated ad the Euclidea distace was used here as the classificatio measure. The results of the classificatio are depicted i Table 3 ad oe ca observe from this table that RALMs achieved higher recogitio rates i the oise case. Table 3 Classificatio results compariso with differet salt-ad-pepper oises. Method Parameter a Noise-free 5% 8% % 5% 2% RALMs 96% 88% 66% 62% 62% 44% 2 % 96% 92% 88% 74% 66% 4 % 98% 92% 94% 78% 76% 6 % 98% 96% 98% 86% 8% 8 % % 96% 98% 8% 84% % % % % 8% 72% 5 % 8% 8% 78% 78% 72% 2 % % 94% 86% 6% 5% OFMMs % % % 88% 84% 74% The fial experimet was aimed at fidig out how well the proposed ivariats perform for color image recogitio. Color images were geerated from the same image dataset as the traiig set [23]. They were cropped ad rotated to a stadard size of pixels. Some samples of these traiig images are give i Figure.The testig images were geerated from the traiig images by rotatig them with rotatio agle θ = 5,, 5,, 36 ad the cotamiatig them with pepper-ad-salt oise with a desity of 2% (see Figure 2). I order to ivestigate the role of the parameter i the recogitio performace, the experimets were repeated for each parameter. The classificatio rates for the OFMMs-based method ad Q-RALMs method are give i Table 4. It was clearly foud that the Q-RALMs had a better classificatio performace at parameter = 6, 8,. Figure Part of color images traiig set.

17 Image Descriptio usig Radial Associated Laguerre Momets 7 Figure 2 Part of color images testig set with 2% pepper-ad-salt oise. Table 4 Classificatio results compariso with differet salt-ad-pepper oises. Method Parameter a Noise-free 5% 8% % 5% 2% RALMs 8% 66% 68% 56% 48% 4% 2 % 94% 82% 74% 74% 62% 4 % 88% 88% 8% 78% 66% 6 % 94% 94% 84% 66% 68% 8 % % % 98% 9% 8% % % % 98% 9% 78% 5 % 84% 78% 76% 68% 62% 2 % 98% 94% 88% 76% 62% OFMMs % % 96% 94% 84% 64% All above the experimets idicated that the parameter plays a importat role i the patter recogitio task whe usig the ivariats of the RALMs-based method, as it cotrols the shiftig to the image regio of iterest. The choice of the parameter correspodig to the case where the emphasis of the momets is at the ceter of the image gave the best recostructio result. The classificatio

18 8 Boju Pa, et al. experimet demostrated that the best recogitio accuracy was achieved for from 6 to uder both oise-free ad oisy coditios. 5 Coclusios This paper itroduced a ew type of orthogoal momets based o the associated Laguerre polyomials for image descriptio, ad costructed RALMs usig methods that are similar to those of disc-based momets. This form makes them particularly suitable for patter recogitio tasks requirig rotatio ivariats. I additio, the study exteded the proposed momets ad rotatio ivariats defied for gray-level images to color images usig the theory of quaterio algebra. The umerical experimet results obtaied from both gray-level images ad color images demostrate that the effectiveess of the proposed ALMs ad RALMs could be better accordig to descriptio performace ad ivariat patter recogitio capabilities i oise-free ad oisy cases. Refereces [] Zhag, H., Shu, H., Ha, G.N., Coatrieux, G., Luo, L. & Coatrieux, J.L., Blurred Image Recogitio by Legedre Momet Ivariats, IEEE Trasactios o Image Processig, 9(3), pp , 2. [2] Jai, S., Papadakis, M., Upadhyay, S. & Azecott, R., Rigid-motioivariat Classificatio of 3-D Textures, IEEE Trasactios o Image Processig, 2(5), pp , 22. [3] Li, Y.H. & Che, C.H., Template Matchig Usig the Parametric Template Vector with Traslatio, Rotatio ad Scale Ivariace, Patter Recogit., 4(7), pp , 28. [4] Yap, P.-T., Jiag, X. & Kot, A.C., Two-Dimesioal Polar Harmoic Trasforms for Ivariat Image Represetatio, IEEE Tras. Patter Aal. & Mach. Itell., 32(7), pp , 2. [5] See, K.W., Loke, K.S., Lee, P.A. & Loe, K.F., Image Recostructio Usig Various Discrete Orthogoal Polyomials i Compariso with DCT, Applied Mathematics ad Computatio, 93(2), pp , 27. [6] Teh, C.H. & Chi, R.T., O Image Aalysis by the Methods of Momets, IEEE Trasactios o Patter Aalysis ad Machie Itelligece, (4), pp , 988. [7] Teague, M.R., Image Aalysis via the Geeral Theory of Momets, Joural of the Optical Society of America, 7(8), pp , 98. [8] Mukuda, R. & Ramakrisha, K.R., Momet Fuctios i Image Aalysis-Theory ad Applicatios, World Scietific, Sigapore, 998.

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